25.10 Complex Hessian equations67A66AB9757E

67A66AB9757E

The cone inequalities and the estimate with a prescribed potential in section 8.1 follow Fang's [v1, Lemma 2.2, Proposition 2.3 and Theorem 3.1, pp. 5–6 and 8–17, FangHessianSmooth26]. The scalar shift in 8.2.3 is proved through real roots, including its endpoint. Polarized positivity then gives the mixed-trace and linearized coefficient bounds. The largest-eigenvalue argument keeps the Chern torsion terms and evaluates every derivative of the Hessian root at the original matrix, while a separate positive perturbation supplies its smooth lower support. In the quadratic weight used for the potential, the bound $-3A\leq \Psi '\leq -A$ determines the choice of the small eigenvalue parameter. The absolute mixed-form estimate below Hessian degree $m$ and the torsion bounds for products are the mechanisms of Kołodziej and Nguyen [arXiv v2, Section 2.1, pp. 4–7, KNHessian16].

The Euclidean Liouville theorem 8.3.2 is due to Dinew and Kołodziej [arXiv v2, Theorem 3.2, pp. 8–11, DKHessian17]. Their rescaling argument is Section 2, pp. 6–7, there; its Hermitian version is also used by Székelyhidi [arXiv v2, Section 6, pp. 26–27, Sze18]. The proof here uses constant-cone linear tests and lower viscosity tests to pass to the maximal limit, so it does not require a variable-background Hessian-product convergence theorem. The averaging proof obtains the uniform directional energy bound with a fixed dependence on the smaller averaging radius. The subsequent local Laplace estimate controls the real Hessian in finite $L^ p$ spaces.

Smooth proper existence and admissible approximation are the mechanisms of Kołodziej and Nguyen [arXiv v2, Theorem 3.17 and Lemma 3.20, pp. 23–25, KNHessian16] and Guedj and Lu [arXiv v1, Theorem 1.7, pp. 4–5, GLHessian25]. Here the smooth trace and gradient estimates first close the proper continuity path. Its penalty equations then approximate every continuous admissible potential uniformly. This construction precedes the definition of continuous Hessian products and does not use a domination theorem. For positive exponential coefficient $a$, rescaling both the metric and the potential reduces the equation to coefficient one. The maximum comparison in 8.8.1 gives the coefficient $a^{-1}$ for continuous data. The existence assertion in 8.8.3 then follows from a contraction on the interval between constant barriers, followed by the continuous regularity theorem.

The decreasing approximation theorem 8.6.6 extends the construction to bounded potentials and recovers Kołodziej and Nguyen's [arXiv v2, Lemma 3.20, pp. 25–26, KNHessian16]. The bounded upper-test characterization is the one used by Guedj and Lu [arXiv v1, Section 1.1.1, pp. 3–4, GLHessian25], through smooth strictly elliptic linear tests. Here successive smooth obstacles are chosen below the preceding admissible approximant, and the penalty error is then made smaller than their fixed gap. This gives a decreasing sequence before defining any product of the bounded potential. The same squeeze proves 8.7.4 after the continuous obstacle envelope has been constructed.

Continuous mixed products and their uniform continuity are [arXiv v2, Proposition 2.11 and Corollary 2.12, pp. 10–11, KNHessian16]. The contact comparison, nonvanishing and domination arguments of section 8.7 follow Guedj and Lu [arXiv v1, Lemma 1.10, Theorems 2.1–2.2 and Corollaries 2.3–2.4, pp. 6 and 8–9, GLHessian25]. The continuous obstacle envelope is constructed by the same penalty solutions, giving continuity and contact support directly. Differentiated background factors are interpreted using the signed lower-degree products. In the mass nonvanishing proof, the telescoping form has degree $n-1$ and retains each power of the original potential. The normalized linear Green estimate rules out an open set on which the stretched difference envelopes tend to $-\infty $.

The bounded recursion in section 8.9 follows the construction of Kołodziej and Nguyen [Lemmas 3.1–3.2, Theorem 3.3 and Proposition 3.7, pp. 125–129, KNHermitianMeasures26]. Their local cone is imposed on $\mathrm{dd}^{\mathrm{c}}u$, whereas the book imposes it on $\omega +\mathrm{dd}^{\mathrm{c}}u$. The proof here therefore expands each raw factor into a shifted admissible factor minus the smooth background before applying the coefficient estimates. The first limit is in the first potential, and the final limit is in the last potential; the top unwedged current remains distributional. The positive shifted wedge has measure coefficients, while its background derivatives are given explicitly in 8.9.7. Permutation invariance does not follow from this ordered construction alone, as the source also points out in [Remark 3.6, p. 128, KNHermitianMeasures26]. Its later capacity and weighted-convergence results concern the source local class. The elementary correction in 8.9.8 places $u+\rho $ in that class while retaining the same Hermitian metric. It preserves the nonclosed background terms and does not itself prove the capacity quasicontinuity needed for higher-degree simultaneous convergence.

The two-factor argument in section 8.10 uses instead the global conformal Gauduchon multiplier of theorem 11.4.2. The energy identity for $u^2$ bounds the Sobolev norm, and first-order integration by parts retains the $\mathrm dT$ term of the nonclosed Gauduchon test. Sobolev truncation and mollification prove the capacity bound for the positive linear trace; the second-cone inequality transfers it to every first-Hessian coefficient. Tail convex combinations from Mazur's lemma identify quasicontinuity with the canonical potential pointwise. A common exceptional set, Dini's theorem and bounded continuous extension then give the weighted convergence used in 8.10.5. This argument controls two factors without applying the source's local capacity results to a shifted cone. The cutoff absolute-coordinate potentials in 8.10.7 show that a two-Hessian measure can charge a smooth real codimension-two set of zero Sobolev capacity. Thus this capacity estimate cannot supply the higher-degree argument.

The local capacity and decreasing-convergence conclusions are studied in [Corollaries 3.18–3.19, p. 137, and Lemma 5.1, pp. 143–145, KNHermitianMeasures26]. In section 8.11, smooth competitors suffice to control the coefficients of the book's ordered currents. A fixed regularized maximum localizes decreasing smooth approximations near the boundary. The smooth energy variation retains every derivative of the Hermitian background and gives convergence in these local capacities. Outer-open coefficient estimates and bounded continuous extension then prove the weighted convergence needed for 8.11.12. This proof uses the global approximation already established in the chapter and avoids an identification of gradients in a singular weighted Sobolev completion. It gives decreasing convergence for the book's shifted class. At degree $m$ the unwedged current converges in distributions, while its positive wedge $Q_ m$ converges as a Radon measure.

The coefficient estimate in 8.2.8 uses Newton's inequality on the list with two coordinates deleted. A related bound appears in [equation (2-8), p. 111, KNHermitianMeasures26]. The torsion estimate at degree $m-1$ corresponds to [Lemma 2.4 and Corollary 2.5, pp. 113–114, KNHermitianMeasures26]. Its proof sums the diagonal wedge coefficients before taking absolute values and compares them with the positive gradient coefficients. This extends the smooth capacity induction to degree $m$. Positive shifted currents are then dominated by the corrected local Hessian traces, and 8.11.14 gives the weighted decreasing convergence studied in [Lemma 5.1(b), pp. 144–145, KNHermitianMeasures26]. The unwedged $D_ m$ retains its distributional meaning.

The strict locality and maximum inequality in 8.12.1, 8.12.2 correspond to [Corollary 5.2, p. 145, KNHermitianMeasures26], which concerns local unshifted Hessian products. The arguments here apply to the book's global shifted products. Continuous functions of decreasing potentials supply the strict-set cutoffs; a decreasing constant shift includes the contact set in the measure inequality.

The torsion-weight identity in 8.12.4 extends the smooth integration-by-parts argument using the coefficient capacity $c_{m-1}$. In 8.12.5, it allows the truncation argument of 8.7.6 to apply to bounded potentials on arbitrary Hermitian backgrounds. This is positivity of each individual mass; it does not supply a uniform positive lower bound over the admissible class.

The bounded mass formula in 8.13.1 extends the integration-by-parts calculation of theorem 11.1.5 using the recursive currents. The necessity argument in 8.13.4 is the polarization mechanism of [Theorems 1–2, Chiose16], used above in theorem 11.1.7. At Hessian degree $m$, only the powers $n-m,\ldots ,n-1$ occur. Sufficiency follows from the mixed maximum inequality, while 8.13.3 keeps the exact total-mass difference on arbitrary backgrounds. The example in 8.13.5 extends the smooth varying-mass torus calculation of example 11.1.12.

The mixed torsion comparison and weak modified comparison in 8.13.7, 8.13.8 use the mechanism of Gu and Nguyen [Lemmas 3.8–3.10, pp. 1197–1200, GuNguyenBoundary18]. Their local bounded-product construction uses a locally conformal Kähler reference metric. Here the global decreasing approximation and weighted torsion currents give the integration-by-parts identity for the chapter's shifted products. A finite mixed-degree induction controls the error by diagonal traces; rescaling the upper potential then absorbs the lower traces on a sufficiently small sublevel set.

The pointwise order in 8.2.10 uses the canonical linear subharmonic representatives of the admissible class. The positive volume term from 8.12.3 and weak modified comparison prove 8.13.10, 8.13.11 on arbitrary Hermitian backgrounds. The same strict comparison proves the logarithmic-data estimate in 8.13.12 for a fixed finite Radon right-hand measure. Together with smooth proper existence, it identifies the bounded solution in 8.13.13 without a separate continuity argument. The continuous-potential domination results in 8.7.7, 8.7.8 use the earlier envelope argument.

The conclusion of 8.8.2 is the continuous-solution case of Fang's [v1, Theorem 4.1, pp. 17–19, FangHessianSmooth26]. The theorem here assumes continuity of the given admissible weak solution; Fang's statement treats bounded solutions and also invokes a continuity result for them. The uniformly elliptic concave extension uses only a bounded complex Hessian. Its regularity input is the continuous-viscosity version of the Evans–Krylov–Caffarelli estimate in Tosatti, Wang, Weinkove and Yang [arXiv v2, Theorem 3.4 and the paragraph following it, pp. 16–20, TWWY15]. It applies directly to the auxiliary limit. The general bounded-solution continuity theory and results for degenerate backgrounds are separate from this argument.